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arXiv · 2511.12211

Grauert's direct image theorem via superconnections and desingularizations

Shu Shen·Jianqing Yu
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Abstract

We give a new differential-geometric proof of Grauert's theorem on the coherence of the higher direct image of a coherent sheaf under a proper holomorphic morphism between complex analytic spaces. In the smooth case, our approach is based on the antiholomorphic superconnection introduced by Block and further developed by Bismut-Shen-Wei. The required finiteness results follow from elliptic theory. In the singular case, we reduce the problem to the smooth setting using Hironaka's desingularization.

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Shu Shen, Jianqing Yu. 2025-11-15. Grauert's direct image theorem via superconnections and desingularizations. https://arxiv.org/abs/2511.12211

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